Question

For regular hexagon \(\mathrm{ABCDEF}\), \(\underline{\mathrm{AB}}+\underline{\mathrm{BC}}+\underline{\mathrm{CD}}+\underline{\mathrm{AF}}+\underline{\mathrm{FE}}+\underline{\mathrm{ED}}=\) (A) \(3 \underline{\mathrm{AD}}\) (B) \(\underline{\mathrm{AD}}\) (C) \(\underline{ }\) (D) \(2 \underline{\mathrm{AD}}\)

Short answer

The sum of the side lengths of the regular hexagon ABCDEF is three times the distance AD. In terms of side length 's', we can write this relationship as \(6s = 3(2s)\). Thus, the correct answer is (A) \(3 \underline{\mathrm{AD}}\).

Step-by-step solution

Set up the problem

First, label the hexagon ABCDEF and note that since it is a regular hexagon, all sides and angles will be equal. Now, let 's' be the side length of the hexagon. We wish to express a relationship between the sum of the sides and the distance AD.

Breaking hexagon into equilateral triangles

Create equilateral triangles by connecting the non-adjacent vertices of the hexagon. When we do this, we'll obtain six equilateral triangles.

Finding the length of AD

We can now concentrate on just one of the equilateral triangles, let's say triangle ACD. Since ABCDEF is a regular hexagon and we have created equilateral triangles, the angles will be 120 degrees which bisects angle CAD. Hence, triangle CAD is a 30-60-90 triangle. Using the ratio of side lengths in a 30-60-90 triangle, which is 1:√3:2, we can find the length of AD. In terms of side length 's', the length of AD is 2s.

Writing the equation

Now we can re-write the question in terms of the side length 's'. It asks for the relationship between the sum of the side lengths and the distance AD. Sum of the side lengths: \(s + s + s + s + s + s \) Distance AD: \(2s\) We can now write the equation: \( 6s = k(2s)\), where k is a constant.

Finding the value of k

Now, let's solve for k by dividing both sides of the equation by 2s: \[\frac{6s}{2s} = k\] \[k = 3\] Now, we have \(6s = 3(2s)\), which means that the sum of the side lengths of the hexagon is three times the distance AD. So, the correct answer is (A) \(3 \underline{\mathrm{AD}}\).

Key concepts

Equilateral Triangles

The magical symmetry of equilateral triangles makes them a fundamental building block in many geometry problems. An equilateral triangle, as the name suggests, is a triangle where all three sides are of equal length. Consequently, all three internal angles are also equal, each measuring 60 degrees.

When it comes to a regular hexagon—which is essentially a six-sided figure with all sides and angles equal—division into equilateral triangles reveals much about its structure. By drawing lines between non-adjacent vertices, a regular hexagon can be neatly partitioned into six equilateral triangles. This dissection not only aids in understanding the hexagon's proportions but also paves the way for solving various mathematical problems associated with it. For example, once a hexagon is split into equilateral triangles, it becomes easier to calculate areas, perimeters, and the relationships between different parts of the hexagon.

30-60-90 Triangle

The 30-60-90 triangle is a special right triangle that has angles measuring 30 degrees, 60 degrees, and 90 degrees. The name itself contains the key to the proportional relationships between the sides; the lengths follow the ratio of 1:√3:2. This means that the side opposite the 30-degree angle is the shortest, often labeled '1', the side opposite the 60-degree angle is √3 times longer, and the hypotenuse (opposite the 90-degree angle) is twice as long as the shortest side.

Using these ratios, mathematicians can solve for unknown side lengths in complex figures. For instance, in the problem involving a regular hexagon ABCDEF, the angle bisector within each equilateral triangle divides it into two 30-60-90 triangles. Thus, applying the side ratio, if the short side of the triangle is the side of the hexagon 's', then the hypotenuse—which is one of the diagonals of the hexagon—will be '2s'. This knowledge is not just useful but crucial to finding missing lengths and understanding the geometry of the figure in question.

Hexagon Side Lengths

Determining the side lengths of a hexagon, especially a regular hexagon, is often straightforward due to its symmetry. In a regular hexagon, all the sides are of equal length. If the side length is known, finding the perimeter is as simple as multiplying the side length by six—as there are six sides.

As we've seen in the example involving regular hexagon ABCDEF, by breaking the hexagon into simpler shapes such as equilateral triangles and using special right triangles like the 30-60-90, we can derive formulas and relationships that describe the hexagon as a whole. The given problem elegantly illustrates how the sum of a regular hexagon’s side lengths is related to its diagonals. Specifically, in this case, knowing one side length 's' allows us to determine that the sum of all side lengths, the perimeter, equals '6s', which can then be used to deduce that it is three times the length of diagonal AD or '3(2s)'. Understanding these relationships facilitates a wide range of calculations about the hexagon.